Computing Novel Multiplicative Zagreb Connection Indices of Metal Organic Networks

Metal organic networks (MONs) are defined as one, two and three dimensional unique complex structures of porous material and subclass of polymer’s coordination. These networks also show extreme surface area, morphology, excellent chemical stability, large pore volume, highly crystalline materials. The major advantages of MONs are tailorability, structural diversity, versatile applications, highly controllable nano-structures and functionality. So, the multi-functional applications of these MONs are made them more helpful tools in many fields of science in recent decade. In this paper, we light on the two different MONs with respect to the number of increasing layers of metal and organic ligands together. We define the novel multiplicative Zagreb connection indices (ZCIs) such that multiplicative fourth ZCI and multiplicative fifth ZCI. We also compute the main results for multiplicative Zagreb connection indices of two different MONs (zinc oxide and zinc silicate).


Introduction
Metal organic networks (MONs) are most popular chemical compounds which consist of metal ions and organic ligands. These networks have large pore diameters, intensive surface areas and giant pore volumes. A variety of MONs is presented in modern chemistry. Therefore, zinc based MONs could be modified into devices for luminescent characteristics, see [1]. The electron-rich T-conjugated fluorescent ligands are friendly to construct Zn based MONs through nucleophilic properties in efficient luminescent sensors, see [2]. MONs are widely used in gas and energy storage devices, assessment of chemicals, separation and purification of different gasesm, sensing, , heterogeneous Copyright © 2021 The Authors. Production and hosting by School of Science, University of Management and Technology is licensed under a Creative Commons Attribution 4.0 International License catalysis, environmental hazard, adsorption analysis, toxicology, biomedical applications and biocompatibility. The cancer imaging, drug delivery and biosensing have been cured with the help of biomedical applications of zinc based MONs. So, the physical stability and mechanical properties of these networks have become a theme of useful content due to the abovementioned specifications.
In non-linear optically active MON + 2 Zn is commonly used as a connecting point to prevail undesired d-d transitions in the visible region. The toxicology, biomedical applications and their biocompatibility are currently reported production procedures of zinc based MONs, see [3]. Eddaoudi et al. [4] discussed the isoreticular series (IRMOF-1 to IRMOF-16) of 16 highly crystalline materials. The fixed and free diameter of pores from IRMOF-1 to IRMOF-16 varies in the range of 3.8-19.1 0 A and 12.8-28.8 0 A , respectively. All the IRMOFs considered as the ordinary topology of ) 13 ( 6
Graph theory provides beneficial tools in the field of moderm chemistry and pharmacology which depict the physical and chemical properties of chemical compounds such as heat of evaporation, flash point, heat of formation, boiling point, melting point, temperature, pressure, tention, partition coefficient, density, and retention in chromatography, see [22][23][24]. The very famous degree based TI was firsly discussed by Gutman and Trinajstić in 1972 to check the chemical physibility for the total π-electron energy of the chemical compound [25]. So, these topological indices (TIs) are one of the tabular (or numeric) tools which show biological, chemical and physical properties of chemical compounds. Awais et al. [26] [31][32][33].
Tang et al. [34] (2009) used the concept of connection number which defined Gutman and Trinajstić in [25] to compute Zagreb connection indices of the subdivision based operations on networks. Nowa-days, these degree and connection number based TIs are abundantely used in the topological properties of four-layered neural networks, see [35]. Javaid et al. [36] and Liu et al. [37] computed these TIs of rhombus silicate & rhombus oxide networks and cellular neural networks. Javaid & Jung [38] and Raheem et al. [39] computed M-polynomial based TIs of silicate & oxide networks and 2Dlattice of three-layered single-walled titania nanotubes. Moreover, Zhao et al. [40] computed reverse degree based TIs of zinc based MONs. Moreover, a variety of networks has been defined by using connection number (or leap degree) based TIs, see [41][42][43][44][45][46][47].
In this paper, we define the multiplicative fourth ZCI and multiplicative fifth ZCI. We also discuss multiplicative first ZCI, multiplicative second ZCI and multiplicative third ZCI. We compute abovementioned multiplicative ZCIs of two different MONs such as zinc oxide (ZNOX (p) =IRMOF-10) and zinc silicate (ZNCL(p)=IRMOF-14) networks with respect to the increasing layers 1 ≥ p , taking both metal nodes and linkers together. The rest of the paper is designed as: section II gives the preliminaries and definitions, Section III gives the main results for different MONs (zinc oxide and zinc silicate) and Section IV gives conclusions.

Preliminaries
The vertex and edge sets are V(G) and E(G) for simple and connected network G. A network is connected if their exists no loops and multiple edges. |V(G)| and |E(G)| are the cardinalities of vertex set and edge set which are equal to u and v, respectively. A path between two vertices generates a connected network. The distance between two vertices m and n is the shortest path between them. It is denoted by ) , ( n m d G . The length of shortest and longest paths between m and n is called m-n geodesic and detour respectively. In general [48], is the open qneighborhood set of n, where q represents a positive integer and is called qdistance degree of a vertex n. In particular , degree of n (number of vertices at distance one from particular vertex n).
connection number of n (number of vertices at distance two from particular vertex n). A network becomes chemical if it holds graphical terms vertex and edge equal to chemical terms atom and bond, respectively. In chemical networks, the degree of any vertex is at most 4. For more chemical terminologies, we suggest to see [49].
Definition 2.2 (see [54]). For a (molecular) network G, the first multiplicative Zagreb index )) ( ( 1 G MZ and second multiplicative Zagreb index )) ( Definition 2.3 (see [55]). For a (molecular) network G, the first multiplicative Zagreb index )) ( These connection based TIs are defined by Ali and Trinajstić [56] (2018). They also reported that these connection based TIs are more correlated among the thirteen physicochemical properties of octane isomers than classical Zagreb indices.
Definition 2.5. For a (molecular) network G, the first multiplicative ZCI )) ( These connection based multiplicative Zagreb indices are defined by Haoer et al. [57]. They used these multiplicative versions by the same sense which named as multiplicative leap Zagreb indices.

Definition 2.7. Zinc Oxide Network (ZNOX(n)):
A chemical compound zinc oxide (ZnO) is insoluble in water which is inorganic compound of white powder shape and density 5.61 g/c 3 m . The zinc oxide is heated with carbon (coke) who reduces to the metal vapor to condense the liquid from which the solid metal freezes.
Zinc is a reactive metal to produce hydrogen gas and zinc ion ( Zn ). It also reduces those metal ions whose reduction potentials are greater than + 2 Zn . Zinc oxide is mostly used in making glazes , rubher, enamels, photoconductive surfaces, pigment in white paint, and protective coating for other metals. Zinc oxide related MON is 3 4 which is also known as IRMOF-10. IRMOF-9 is a catenated version of IRMOF-10. IRMOF-10 is three dimensional cubic networks with pore size 16.7/20.2 0 A in diameter, see [58]. The MON of zinc oxide of dimention 1 is presented in Figure 1. Let ≅ R ZNOX(p) be the metal organic (zinc oxide) network of dimention p in the plane, see Figure 1. The partition of R with respect to vertex set V( R ) and edge set E( R ). We can easily see that each vertex of degrees and connection numbers sets are { } , where | V1| = 2p+6, | V2| = 28p+20, | V3| = 30p+10, | V4| = 8p+8 and | V5| = 2p+2. So, |V ( R )| = | V1| + | V2| + | V3| + | V4| + | V5| = 70p + 46. Now, the partition of vertices with respect to degrees and connection numbers are Definition 2.8. Zinc Silicate Network (ZNSL(n)): Silicate ) ( 4 SiO is the most wonderful class of minerals. Silicate is the chemical mixture of metal carbonate or metal oxide with sand. Tetrahedra is used as the basic unit of silicate. So, all silicates gain 4 SiO tetrahedral. In chemistry, silicon ions and oxygen ions are represented by the centre vertices and corner vertices of 4 SiO respectively. In graph theory, we show centre vertices and corner vertices of 4 SiO with silicon nodes and oxygen nodes. If we require a variety of silicate networks, it is easy to change the arrangement of the tetrahedron silicate. Zinc silicate related MON is which is also known as IRMOF-14. IRMOF-14 is three dimensional cubic structures with pore size 14.7/20.1 0 A in diameter, see [58]. The MON of zinc silicate of dimention 1 is presented in Figure 2.

Main Results for MONs
In this section, we compute the main results for first multiplicative ZCI, second multiplicative ZCI, third multiplicative ZCI, fourth multiplicative ZCI and fifth multiplicative ZCI of two different MONs (zinc oxide and zinc silicate).

Conclusions
We computed multiplicative Zagreb connection indices such as first multiplicative ZCI, second multiplicative ZCI, third multiplicative ZCI, fourth multiplicative ZCI and fifth multiplicative ZCI of two different MONs which are zinc oxide (R) and zinc silicate (S) networks with respect to the increasing layers 1 ≥ p , taking both metal nodes and linkers together. Now, the problem is still open for product, subdivision, prism and their compliment networks with the help of degree as well as connection number indices.